Problem: $f(x, y, z) = \sin(z) - xe^y$ What is the Laplacian of $f(x, y, z)$ ? $\nabla^2 f = $
Explanation: The Laplacian of a scalar field $f$ is the sum of each of its second partial derivatives. $\nabla^2 f = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$ [What does it mean to square the gradient?] Let's find the second partial derivatives of $f$ ! $\begin{aligned} f_{xx} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right] = \dfrac{\partial}{\partial x} \left[ -e^y \right] = 0 \\ \\ f_{yy} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial y} \right] = \dfrac{\partial}{\partial y} \left[ -xe^y \right] = -xe^y \\ \\ f_{zz} &= \dfrac{\partial}{\partial z} \left[ \dfrac{\partial f}{\partial z} \right] = \dfrac{\partial}{\partial z} \left[ \cos(z) \right] = -\sin(z) \end{aligned}$ The Laplacian is $\nabla^2 f = f_{xx} + f_{yy} + f_{zz}$. Therefore: $\nabla^2 f(x, y, z) = -\sin(z) - xe^y$